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\title{Non eqilibrium steady states for weakly reversible chemical reaction networks.}
\author{}
\date{\today}

\begin{document}
\maketitle

\begin{frame}
  \frametitle{The notation} 
  Given: 

  \begin{itemize}
	\item A set $\#\mathcal{S}=m$ of species.
	\item A set $\#\mathcal{C}=n$ of complexes.
	\item A directed reaction graph $G(\mathcal{C},E)$ 
		 (where an edge $i\rightarrow j$ if complex $i$ reacts to form complex $j$).
	 \end{itemize}
	 Let
	 \begin{itemize}
	   \item $Y_{i,j}$ be the stoichiometry of species $i$ in complex $j$.
	   \item $A_k = A^T - D = A^T -diag(A\1)$ be the transpose of the graph Laplacian.
	   \item $\psi_j (c) = \prod_i {c_{i}}^{Y_{i,j}}$ be the potential of complex $j$ given the species concentrations.
	 \end{itemize}
  \end{frame}
  \begin{frame}
	\frametitle{The problem}
	Given the dynamics of the network \[\dot{c}=YA_k\psi(c) - b.\]
	Does there exist a concentration $c$ such that $\dot{c}=0$?

	We can write \begin{eqnarray}
	  YA_kv = b, \notag
	  \\ \psi(c) = v. \notag
	\end{eqnarray} If $c>0$, this implies $Y^T\log(c) = \log(v)$.
  \end{frame}
  \begin{frame}
	\frametitle{Our previous result}
	If $G$ is formed by a weakly connected component, the reactions 
	are mass conserving and $b=0$, then there exists $c>0$ such that
	\[YA_k\psi(c) = 0.\]
	{\bf A note on mass conservation:}
	If the reactions are mass conserving they use and produce the same mass.
	If there is a reaction $i\rightarrow j$, then $i$ and $j$ have
	the same mass. All complexes in the same connected component have the same mass.
  \end{frame}
  \begin{frame}
	\frametitle{Some Conjectures}
	\begin{itemize}
	  \item	All networks which can be partitioned into weakly reversible subnetworks have
	steady states for $b=0$.
\item If $b$ is in the range of $YA_k$, and the above is true, then there is a concentration
  that achieves $YA_k\psi(c) = \alpha b$ for some $\alpha>0$. 
 \end{itemize}
  \end{frame}

  \begin{frame}
	\frametitle{A new result}
	If the network is weakly reversible \footnote{Formed by a single terminal-linkage class}, then for some $c>0$, \[YA_k\psi(c) = b.\]

	{\bf Some definitions:} 
	Since $b$ is in the range of $YA_k$, we can write $b= YA_k\eta$.
	Therefore we can select $0<s\in \Re^n$ such that \[\eta^- = \eta + s \textmd{ and } \eta^+=\eta + D^{-1}A^Ts,\]
	with $b=YA^T\eta^+-YD\eta^-$ and $\eta^+,\eta^- > 0$.
  \end{frame}
  \begin{frame}
	\frametitle{Proof strategy}
	We form the problem 
	\begin{align}
	  \min v^TD(\log(v)-1) + v_0(\log(v_0) -1) \notag
	  \\ \textmd{Subject to:} 
	  \\ YDv+YA^T\eta^+v_0 = YA^T\gamma+YD\eta^-\gamma_0 \notag
	  \\ v\geq 0 \notag.
	  \label{P0}
	\end{align}
	This is strictly convex, parametrized by $(\gamma,\gamma_0)$. Therefore
	we can define $(\gamma,\gamma_0)\rightarrow(v^\star,v_0^\star)$. Using mass conservation
	we show that this mapping has a fixed point $(v^\star,\alpha)$.
  \end{frame}

  \begin{frame}
	\frametitle{At the fixed point}
The linear constraint becomes \[ YA_k\hat{v} = YA^T\hat{v}-YD\hat{v} = YA^T\eta^+\alpha_0 - YD\eta^-\alpha_0 = \alpha_0b, \] and
if it is positive, form the KKT conditions 
	and $Y^T\lambda = \log(\hat{v}).$

	We must now investigate. When is the fixed point positive?
\begin{lemma}
The minimizer of the entropy-like functional will have support at least as large as any feasible point.
\end{lemma}
  \end{frame}

  \begin{frame}
	\frametitle{Using the lemma}

	Let $\hat{v},\hat{v_0}$ be the fixed point, there 
	\[ YD\hat{v}+YA^T\eta^+\hat{v_0} = YA^T\hat{v}+YD\eta^-\hat{v_0} \]
	\begin{itemize}
	  \item If $\hat{v_0}>0$ then $(D^{-1}A^T\hat v+\eta^-\hat{v_0},0)$ is feasible, and a convex 
		combination \[((1-\alpha)(D^{-1}A^T\hat{v}+\eta^-\hat{v_0}),\alpha \hat{v_0})\] is strictly feasible.
	  \item Assume $\hat{v_0}=0$ but there is a $\hat{v}_j>0$ then \[(D^{-1}A^T\hat v,0)\] is feasible.
		Therefore $supp(\hat v)\supset supp(A^T\hat v)\cup\sup(\hat v)$. This spreads to all the connected component. 
		And we can use that to show that $\hat{v_0}>0$. 
	\end{itemize}
  \end{frame}
  \end{document}

